Integral Values
⎩ ⎪ ⎨ ⎪ ⎧ z x = y 2 x 2 z = 2 ( 4 x ) x + y + z = 1 6
Find the integral values of x , y , and z satisfying the system of equations above. Submit x y z .
The answer is 108.
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1 solution
Can you please explain how z x = y 2 x implies z = y 2 . You are assuming that z is positive ?
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y 2 ≥ 0 is non-negative, hence z = y 2 ≥ 0 .
Given ⎩ ⎪ ⎨ ⎪ ⎧ z x = y 2 x 2 z = 2 ( 4 x ) x + y + z = 1 6 . . . ( 1 ) . . . ( 2 ) . . . ( 3 )
From ( 2 ) : 2 z = 2 ( 4 x ) = 2 2 x + 1 , ⟹ z = 2 x + 1 .
From ( 3 ) : x + y + z = x + y + 2 x + 1 = 1 6 , ⟹ y = 1 5 − 3 x .
From ( 1 ) :
z x 2 x + 1 2 x + 1 9 x 2 − 9 2 x + 2 2 4 ( 9 x − 5 6 ) ( x − 4 ) ⟹ x y z = y 2 x = ( 1 5 − 3 x ) 2 = 2 2 5 − 9 0 x + 9 x 2 = 0 = 0 = 4 = 2 x + 1 = 9 = 1 5 − 3 x = 3 Taking only the integer solution
Therefore x y z = 4 × 9 × 3 = 1 0 8 .